Lemma 6.2 (Lemma 2.1(ii) in [1]}) 
Let \(\tilde{D}:[0,\infty )\rightarrow [0,\infty )\)  be a monotonic non-decreasing function that satisfies the following (Log-Log Lipschitz) condition: there is \(r>0\)  such that for every \(x>0\)  and \(\Delta >1\)  we have
\(\tilde{D}(\Delta x)\le \Delta ^{r}\tilde{D}(x).\) 
